Fixed Point Theory in Modular Function Spaces by Mohamed A. Khamsi & Wojciech M. Kozlowski

Author:Mohamed A. Khamsi & Wojciech M. Kozlowski
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where f 0 is a fixed function and is Lebesgue measurable. For the kernel k we assume that(a) is Lebesgue measurable

(b)

(c) is continuous, convex and increasing to

(d) for and .

Assume, in addition, that for almost all and all measurable functions there exists a constant such that

Setting and using Jensen’s inequality it is easy to show that ρ is a function modular and that on , that is, T is K-Lipschitzian with respect to ρ. Let us summarize what we have done: given an integral operator we have constructed a modular function space in which this operator is Lipschitzian with the constant K. Obviously, if then T becomes a ρ-contraction, or if , ρ-nonexpansive mapping. As we will see, applying relevant fixed point theorems one can solve the corresponding Urysohn integral equation.

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